Answer:
Let's assume the height of the cylinder to be 'h' and its radius to be 'r'. We can use the given information to form two equations.
First, we know that the volume of the cylinder is 1540 cm³:
V = πr²h = 1540
Second, we know that the difference between the height and the radius is 3 cm:
h - r = 3
Using the second equation, we can solve for 'h' in terms of 'r':
h = r + 3
Substituting this into the first equation, we get:
πr²(r + 3) = 1540
Expanding the left side of the equation and simplifying, we get:
πr³ + 3πr² - 1540 = 0
We can solve for 'r' using the cubic formula, but it's easier to notice that 'r' must be close to 10. We can try different values of 'r' until we get an answer close to 1540. If we try 'r' = 10, we get:
π(10)²(10+3) = 3300π
So, the radius is approximately 10 cm and the height is approximately 13 cm.
To find the total surface area of the cylinder, we need to find the area of the top and bottom circles and the area of the curved surface. The area of each circle is πr², so the total area of the circles is:
2πr² = 200π cm²
The area of the curved surface is the circumference of the circle times the height of the cylinder, or 2πrh. Substituting the values of 'r' and 'h' that we found earlier, we get:
2π(10)(13) = 260π cm²
Therefore, the total surface area of the cylinder is:
200π + 260π = 460π ≈ 1442.4 cm²
So, the total surface area of the cylinder is approximately 1442.4 cm².